# Solution for $x=e^x$ in terms of elementary functions

While solving a problem I've come across an expression which essence is

$$x=e^x$$

It does contain more terms, but I couldn't even figure out an explicit expression for $x$ in terms of elementary functions, much less for the one that I actually came across, which is of the form

$$x=\alpha+\beta e^{\gamma x}$$

Is it possible to express the solution of this equation for $x$ in terms of elementary functions? Any ideas?

• Lambert W-Function - no elementary functions. You can use numerical methods. – Moo Nov 17 '16 at 20:54
• Depends on what you mean by "elementary functions". If you allow Lambert's W function, this is possible. If you restrict yourself to a tigher range of allowed functions, like only exponentials and logarithms, then the answer is conjecturally "there is no elementary expression", which AFAIK follows from Schanuel's conjecture. – Wojowu Nov 17 '16 at 20:58
• I see, it seems indeed this function does not qualify for what I was looking for, so I guess this closes the discussion. Thank you both for your help. – F.Webber Nov 17 '16 at 20:58
• Of most use, try the general solution – Simply Beautiful Art Nov 17 '16 at 20:59
• Oh, thanks @SimpleArt, I guess I'll have to settle with expressing it like this. – F.Webber Nov 17 '16 at 21:03

If the expression $x=e^x$ has a solution, it is simply a number $x_0$, not a function.

Note that $x=e^x \implies (-x)e^{(-x)}=-1$. The Lambert W function, $W(z)$, is defined as

$$z=W(z)e^{W(z)}$$

Therefore, $x=e^x$ is equivalent to $x=-W(-1)$.

Note that inasmuch as $e^x\ge x$ for all $x$, the solution $x=-W(-1)$ is not a purely real number.

• How do you make the conclusion "The solution is not possible in terms of elementary functions"? – Wojowu Nov 17 '16 at 21:10
• @Wojowu First sentence, last paragraph of the Wikipedia before the contents box. – Simply Beautiful Art Nov 17 '16 at 21:11
• @Wojowu The solution is a complex number so it is possible to write it in terms of the value of an elementary function. For example, let $f(z)=-W(-1)z$. Then, $f(1)$ is the solution. So, I've edited and removed the claim. – Mark Viola Nov 17 '16 at 21:17
• @SimpleArt The solution is simply a number, not a function. So to discuss solutions in terms of elementary functions is subject to question. – Mark Viola Nov 17 '16 at 21:21
• Thanks for the discussion, it solves my doubt. I apologize for being a bit vague in the description as I'm noticing now the simplification I made allowed room for ambiguity, because in fact, the expression I come across does involve variables instead of numbers, and the $x$ I'm looking for is a function of such variables. In any case, the answer to my question is very much negative. – F.Webber Nov 17 '16 at 21:23